Integrand size = 10, antiderivative size = 46 \[ \int \frac {\text {Chi}(a+b x)}{x^2} \, dx=\frac {b \cosh (a) \text {Chi}(b x)}{a}-\frac {b \text {Chi}(a+b x)}{a}-\frac {\text {Chi}(a+b x)}{x}+\frac {b \sinh (a) \text {Shi}(b x)}{a} \]
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Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6668, 6874, 3384, 3379, 3382} \[ \int \frac {\text {Chi}(a+b x)}{x^2} \, dx=-\frac {b \text {Chi}(a+b x)}{a}-\frac {\text {Chi}(a+b x)}{x}+\frac {b \cosh (a) \text {Chi}(b x)}{a}+\frac {b \sinh (a) \text {Shi}(b x)}{a} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 6668
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Chi}(a+b x)}{x}+b \int \frac {\cosh (a+b x)}{x (a+b x)} \, dx \\ & = -\frac {\text {Chi}(a+b x)}{x}+b \int \left (\frac {\cosh (a+b x)}{a x}-\frac {b \cosh (a+b x)}{a (a+b x)}\right ) \, dx \\ & = -\frac {\text {Chi}(a+b x)}{x}+\frac {b \int \frac {\cosh (a+b x)}{x} \, dx}{a}-\frac {b^2 \int \frac {\cosh (a+b x)}{a+b x} \, dx}{a} \\ & = -\frac {b \text {Chi}(a+b x)}{a}-\frac {\text {Chi}(a+b x)}{x}+\frac {(b \cosh (a)) \int \frac {\cosh (b x)}{x} \, dx}{a}+\frac {(b \sinh (a)) \int \frac {\sinh (b x)}{x} \, dx}{a} \\ & = \frac {b \cosh (a) \text {Chi}(b x)}{a}-\frac {b \text {Chi}(a+b x)}{a}-\frac {\text {Chi}(a+b x)}{x}+\frac {b \sinh (a) \text {Shi}(b x)}{a} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {\text {Chi}(a+b x)}{x^2} \, dx=\frac {b x \cosh (a) \text {Chi}(b x)-(a+b x) \text {Chi}(a+b x)+b x \sinh (a) \text {Shi}(b x)}{a x} \]
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\[\int \frac {\operatorname {Chi}\left (b x +a \right )}{x^{2}}d x\]
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\[ \int \frac {\text {Chi}(a+b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x + a\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\text {Chi}(a+b x)}{x^2} \, dx=\int \frac {\operatorname {Chi}\left (a + b x\right )}{x^{2}}\, dx \]
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\[ \int \frac {\text {Chi}(a+b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x + a\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\text {Chi}(a+b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x + a\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\text {Chi}(a+b x)}{x^2} \, dx=\int \frac {\mathrm {coshint}\left (a+b\,x\right )}{x^2} \,d x \]
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